Online Non-Monotone DR-Submodular Maximization Matching the Offline $0.401$ Factor

• URL: http://arxiv.org/abs/2609.02145v1
• Date: Wed, 02 Sep 2026 06:00:10 GMT
• Title: Online Non-Monotone DR-Submodular Maximization Matching the Offline $0.401$ Factor
• Abstract summary: We study asymmetric non-negative, non-monotone DR-submodular functions over compact convex subsets of the $d$-oracle unit.<n>Online algorithm attains factor 0.401$ with sub approximate regret when feedback is conditionally unbiased and bounded.
• Score: 47.30196146041476
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $δ\in[0,1/4]$, batching gives $O(T^δ)$ calls per round and $O(T^{4/5-δ/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.

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